Supports of doubly stochastic measures

Abstract

Recent work has shown that extreme doubly stochastic measures are supported on sets that have no axial cycles. We give a new proof of this result and examine the supporting set structure more closely. It is shown thai the property of no axial cycles leads to a tree like structure which naturally partitions the support into a collection of disjoint graphs of functions from the jr-axis to the y-axis and from the y-axis to the x-axis. These functions are called a limb numbering system. It is shown that if the disjoint graphs in the limb numbering system are measurable, then the supporting set supports a unique doubly stochastic measure. Further, the limb structure can be used to develop a general method for constructing sets which support a unique doubh stochastic measure. © 1995 Chapman & Hall.